Applications of definite integral

The function $f(x)=\sqrt{x}$ is graphed in the picture. Consider the region bounded by the graph of $f$ on $[1;4]$, lines $x=1$, $x=4$ and the $x$-axis. Identify the formula for volume of the solid of revolution obtained by revolving this region about the $x$-axis.

Part of the graph of the function $f(x)=\frac{1}{x}$ is shown in the picture. Complete the following sentence: „Formula $$V=\pi {\int }_1^2{x}^{-2}\mathrm{d}x$$ determines the volume of the solid of revolution obtained by revolving region bounded by ...”

Consider a conic section defined by $x^2-4y^2=1$. Find the volume of the solid obtained by rotating the given conic about the $x$-axis on the interval $[1;6]$.

Consider a conic section defined by $y^2=x+a$. Find the value of the real parameter $a$, if a solid of the volume $\frac{125\pi }{2}$ is obtained by rotating the given conic about the $x$-axis on the interval $[0;5]$.